Finite Group Databasebeta

← #10876198

Subgroups — #10876198

Order 768.

872 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82228C211yesmaximal, characteristic
2#73863C211yesmaximal, characteristic, complemented
3C3×#1854C211yesmaximal, characteristic, complemented
4#71576C211yesmaximal, characteristic, complemented
5#70640C211yesmaximal, characteristic, complemented
6#70624C211yesmaximal, characteristic, complemented
7#70591C211yesmaximal, characteristic, complemented
8#70569C211yesmaximal, characteristic, complemented
9#69960C211yesmaximal, characteristic
10#69910C211yesmaximal, characteristic, complemented
11#69522C211yesmaximal, characteristic
12#68915C211yesmaximal, characteristic, complemented
13#68903C211yesmaximal, characteristic, complemented
14#68793C211yesmaximal, characteristic, complemented
15#68791C211yesmaximal, characteristic, complemented
16#27590—31nomaximal, Sylow
17C22×#235C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C4×Dic3C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C3×#380C2211yescharacteristic
24#5212C2211yescharacteristic
25C2×#830C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30#4988C2211yescharacteristic
31#4988C2211yescharacteristic
32#4977C2211yescharacteristic
33#4977C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4958—21no—
45#4958—21no—
46#4943—21no—
47Dic3×C22⋊C4—21no—
48C4×(C2×D3)⋊C4—21no—
49C4×(C2×D3)⋊C4—21no—
50#4651—21no—
51#4648—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2297—31no—
61#2283—31no—
62#2097—31no—
63#2092—31no—
64#2074—31no—
65#2073—31no—
66#1928—31no—
67#1927—31no—
68#1926—31no—
69#1925—31no—
70#1854—31no—
71#1854D311yescharacteristic, complemented
72#1817—31no—
73#1788—31no—
74#1788—31no—
75C23×C3×C4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3D411yescharacteristic
78C23×Dic3D411yescharacteristic
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235—22no—
83C2×#235—41no—
84C2×#235—41no—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×Dic6C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C22×C3×C8C2311yescharacteristic
90C3×#117—41no—
91C3×#117—41no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C42—21no—
99C2×#217—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4×Dic3—22no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3D411yescharacteristic
111C2×C4×Dic3D411yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C3×#107—41no—
117C3×#106—41no—
118C3×C22⋊C8—22no—
119C3×#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#830—22no—
132#821—22no—
133#821—41no—
134#816—22no—
135#816—41no—
136#810—22no—
137#810—41no—
138#809—22no—
139#809—41no—
140#808—41no—
141#807—41no—
142C22×#41—31no—
143C2×#117C2×D311yescharacteristic
144C22×C42—31no—
145C2×#107—31no—
146C2×#107C2×D311yescharacteristic
147C2×#106—31no—
148C2×#106C2×D311yescharacteristic
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×C22⋊C8—31no—
158C2×C22⋊C8C2×D311yescharacteristic
159#391—31no—
160#391—31no—
161#390—31no—
162#390—31no—
163#386—61no—
164#386—61no—
165#383—61no—
166#382—31no—
167#382—31no—
168#381—61no—
169#380—31no—
170#380C2×D311yescharacteristic
171C4×C4⋊C4—31no—
172C4×C4⋊C4—31no—
173C4×C22⋊C4—61no—
174C4×C22⋊C4—61no—
175C4×C22⋊C4—61no—
176#376—61no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C24×C3C2×D411yescharacteristic
186C23×D3—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4#4112yes—
197C22×C3×C4#4112yes—
198C22×C3×C4C2411yescharacteristic
199C2×#67—24no—
200C2×#67—42no—
201C2×#67—42no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—41no—
209C22×Dic3—41no—
210C22×Dic3—41no—
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215#235—44no—
216#235—44no—
217#235—44no—
218#235—44no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12#4112yes—
225C2×C4×D3—24no—
226C2×C4×D3—42no—
227C2×C4×D3—42no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6#4112yes—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—42no—
250C3×C42—42no—
251#217—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4×Dic3—22no—
261C4×Dic3—22no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×#41—62no—
275C2×#41—62no—
276C2×#41—62no—
277C2×#41—62no—
278C2×#41—121no—
279C2×#41—121no—
280C22×Q8—31no—
281C22×D4—31no—
282C22×D4—61no—
283C23×C4—31no—
284C23×C4—31no—
285C23×C4—61no—
286C23×C4C22×D311yescharacteristic
287C22×C8—31no—
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C22×C8C22×D311yescharacteristic
291#117—41no—
292#117—41no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—21no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—62no—
312C2×C42—62no—
313C2×C42—61no—
314C2×C42—61no—
315#107—41no—
316#107—121no—
317#106—41no—
318#106—121no—
319#103—62no—
320#103—62no—
321#103—121no—
322#103—121no—
323#102—62no—
324#102—62no—
325#102—121no—
326#102—121no—
327C22⋊C8—22no—
328C22⋊C8—62no—
329#95—21no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3#12312yes—
348C23×C3C22×D411yescharacteristic
349C22×D3—22no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C22×D3—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4#12312yes—
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381#67—48no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—22no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—44no—
395C2×Dic3—44no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3#13512yes—
405D12—22no—
406D12—22no—
407D12—41no—
408D12—41no—
409C4×D3—48no—
410Dic6—22no—
411Dic6—22no—
412Dic6—41no—
413Dic6—41no—
414C3×C8—41no—
415C3×C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—61no—
419C24D3×D411yescharacteristic
420#41—124no—
421#41—124no—
422#41—124no—
423#41—124no—
424C2×Q8—32no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×D4—32no—
431C2×D4—64no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—122no—
438C2×D4—122no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—64no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—122no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4—122no—
463C22×C4#23512yes—
464C22×C4D3×D411yescharacteristic
465C22×C4#23912yes—
466C22×C4C23×D311yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—122no—
494C22⋊C4—122no—
495C22⋊C4—122no—
496C22⋊C4—122no—
497C22⋊C4—122no—
498C42—42no—
499C42—62no—
500C42—62no—
501C42—61no—
502C42—61no—
503C42—61no—
504C42—61no—
505C42—122no—
506C42—122no—
507C22×C3—22no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3C2×#12311yescharacteristic
515C22×C3#52511yescharacteristic
516C22×C3#53511yescharacteristic
517C22×C3C2×#13511yescharacteristic
518C22×C3#57511yescharacteristic
519C22×C3#57511yescharacteristic
520C22×C3#57511yescharacteristic
521C2×D3—42no—
522C2×D3—42no—
523C2×D3—42no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—81no—
527C2×D3—81no—
528C3×C4—22no—
529C3×C4—81no—
530C3×C4—81no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4#55111yescharacteristic
534C3×C4#55111yescharacteristic
535C3×C4#55111yescharacteristic
536C3×C4#55111yescharacteristic
537Dic3—22no—
538Dic3—41no—
539Dic3—41no—
540Dic3—41no—
541Dic3—42no—
542Dic3—81no—
543Dic3—81no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—62no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23—121no—
555C23—121no—
556C23—121no—
557C23#89412yes—
558C23C2×D3×D411yescharacteristic
559Q8—62no—
560Q8—62no—
561Q8—121no—
562Q8—121no—
563D4—62no—
564D4—62no—
565D4—128no—
566D4—121no—
567D4—121no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—32no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—62no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—61no—
598C2×C4—61no—
599C2×C4—128no—
600C2×C4—124no—
601C2×C4—124no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4#89412yes—
611C2×C4C2×#23511yescharacteristic
612C2×C4#100411yescharacteristic
613C2×C4#100411yescharacteristic
614C2×C4C2×#23911yescharacteristic
615C2×C4D3×#4111yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C8—41no—
618C8—41no—
619C8—121no—
620C8—121no—
621C2×C3—42no—
622C2×C3#248211yescharacteristic
623C2×C3#309111yescharacteristic
624C2×C3#309311yescharacteristic
625C2×C3#310211yescharacteristic
626C2×C3#311111yescharacteristic
627C2×C3#321211yescharacteristic
628C2×C3#321911yescharacteristic
629D3—82no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—122no—
638C22—122no—
639C22—122no—
640C22—241no—
641C22—241no—
642C22—241no—
643C22—241no—
644C22C2×#89411yescharacteristic
645C22#559211yescharacteristic
646C22#565611yescharacteristic
647C22D3×#13511yescharacteristic
648C22#576811yescharacteristic
649C22#577311yescharacteristic
650C22#577711yescharacteristic
651C4—22no—
652C4—62no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—121no—
658C4—121no—
659C4—121no—
660C4—122no—
661C4—241no—
662C4—241no—
663C4#555611yescharacteristic
664C4#555611yescharacteristic
665C4#563711yescharacteristic
666C4#563711yescharacteristic
667C3#2759011yescharacteristic, complemented, Sylow
668C2—42no—
669C2—242no—
670C2#7461411yescharacteristic
671C2#7918511yescharacteristic
672C2#7921611yescharacteristic
673C2#8007911yescharacteristic
674C2#8022111yescharacteristic
675C2#8134811yescharacteristic
676C2#8149411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1854C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1854D3
hypocenterC3#27590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #73863C2semidirect
3. C3×#1854C2semidirect
4. #71576C2semidirect
5. #70640C2semidirect
6. #70624C2semidirect
7. #70591C2semidirect
8. #70569C2semidirect
10. #69910C2semidirect
12. #68915C2semidirect
13. #68903C2semidirect
14. #68793C2semidirect
15. #68791C2semidirect
71. #1854D3semidirect
667. C3#27590semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#822282
#738632
C3×#18542
#715762
#706402
#706242
#705912
#705692
#699602
#699102
#695222
#689152
#689032
#687932
#687912
#275903